Intrinsic equation: .Cartesian parametrization: where
is the parametrization of a curve with constant curvature (a skew
circle), and
is the parametrization of a curve with constant torsion.

Bertrand curves are the 3D curves the curvature and torsion
of which are linked by an affine non linear relation (hence the above intrinsic
equation) - the linear case yields the helices.

A curve
is a Bertrand curve if and only if there exists a curve
different from
with the same principal normal line as .

Except for the case of a circular helix, the curve
is unique; the distance between two corresponding points along the common
normal line is constant, and the angle formed by the corresponding tangents
is constant.

Examples: the circular helix and, more generally, the
skew
circles (case where
b = 0; the angle formed by the tangents
is then a right angle and each curve is the locus of the centers of curvature
of the other).